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Modulation analysis of phase-shifted holographic interferograms
Optics Communications 105 (1994) 379-387 North-Holland
OPTICS COMMUNICATIONS
Full length article
Modulation analysis of phase-shifted holographic interferograms X.-Y. Su 1, A.M. Z a r u b i n a n d G. y o n Bally Laboratory of Biophysics, Institute of Experimental Audiology, Universityof Miinster, Kardinal-von-Galen-Ring 10, D-48129 Miinster, Germany Received 7 April 1993; revised manuscript received 20 September 1993
A new evaluation technique for phase-shiftedholographic interferogramsis developed and verified by experiments. In evaluating the interferograms,both the discrete phase distribution and the intensity modulation of interference fringes are calculated and a binary control mask is generatedby analysinghistogramsand tracing local minima of the intensity modulation distribution. The control mask is used to identify valid and invalid areas of the discrete phase distribution in subsequent phase unwrapping and phase interpolation, and to control the path of phase unwrapping. The evaluation of experimentallyobtained phase-shifted holographicinterferograrnsshowsthat this method can be successfullyapplied for the processingof complexinterferencepatterns, even if the interferogramshave fringe discontinuities and shaded areas.
1. Introduction
Holographic interferometry is a powerful technique of deformation and vibration analysis in biomedical and engineering applications [ 1-3 ]. With the development of high resolution CCD camera and the availability of PC-based image grabber, digital holographic interferometry ( D H I ) has become an important diagnostic tool in the quantitative experimental investigations [4-6]. The D H I based on phase shifting technique permits parallel acquisition and processing of large amounts of data with high accuracy, because the phase shifting holographic interferometry utilises a temporal sequence of full-field interferograms of the same phase distribution with fringe shifts within period of the fringes. The phase mapping from digitised N frame fringe data to the wrapped phase is a point to point operation which does not require the analysis of the surrounding points, so there is no problem of spectra overlapping in the spatial frequency domain, which often happens in D H I based Fourier fringe analysis. The later has an obvious advantage that only one interferogram is required, but this advantage can not be rePresent address: Opto-ElectronicsDepartment, Sichuan University, Chengdu 610064, P.R. China.
alised with complex fringe patterns because of the spectra overlapping. When phase-shifting holographic interferometry is applied for deformation measurement of the complex objects, there still are some problems to be solved. First, physical discontinuities of the surface of the complex objects often result in fringe discontinuities in the interferogram. Second, physical discontinuities of the object surface cause local shadows in the interferograms. In the areas of the local shadows the fringes disappear and the phase has an uncertain value. In the areas of the fringe discontinuities the phase changes rapidly. If the phase variation between neighbouring sampling points is more then n, the phase unwrapping problem appears. The third problem is the separation of the measured object from the background. These problems have been the obstacles to the widespread application of holographic interferometry for automated quantitative analysis of the complex objects. In order to control the quality of evaluation of the phase-shifted interferograms and, in particular, to distinguish true interference fringes and other structures such as shadows, contours, holes etc., a proposal has been made [ 7 ] to use the fringe contrast distribution and the mean intensity distribution. However, no algorithm incorporating this proposal
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0030-4018 (93)E0487-Z
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and the phase unwrapping ha:. been described, and no experimental results have been presented. The phase unwrapping problem has been extensively studied because of its importance in interferometry and profilometry, but as noted in ref. [8 ], the task of phase unwrapping is difficuL and has been worked on by many people without I5 ~ding an entirely satisfactory solution. Recently, se, reral methods [9-11 ] have been suggested and realised experimentally which cope with interferogran~ evaluation to a great extent. However considerable difficulties still remain when the situation is mare complex. For example, the linkage between a r air of poles [ 9 ] must not be, in general, a straight 1me. Problems appear also if there is an integer fringe shift across the discontinuity, or the spatial frequencies on either side of the discontinuity are not sufficiently different. These situations often occur in the deformation field of the complex objects due to Lhe surface discontinuities. Therefore, in this case a more advanced algorithm is required. In paper [12], a new method of phase-shiffing fringe pattern analysis appliec to profilometry and the experimental results for the 3D shape measurement of a complex object has been described. The purpose of this paper is to de:;cribe the application of the method in D H I and to present experimental results of the complex deforxmtion measurement based on the analysis of hologlaphic interferograms. We introduce into the evaluation process an additional parameter, the intensity modulation of the interference fringes (which is di:;tinct, both physically and mathematically, from the fringe contrast parameter [ 7 ] ), and calculate both lhe discrete phase and intensity modulation distribwJon using N sampled phase-shifted frames of the int,~rference pattern. The following analysis shows thal this parameter contains important information o:a the accuracy and reliability of the discrete p h a s e the physical discontinuities in the fringe pattera, and the boundary between the measured object and the background. By analysing a histogram of the intensity modulation distribution in the whole pat:~ern and tracing local minima of the modulation distribution, a binary control mask is generated. Thq; control mask is used to identify valid and invalid areas of the discrete phase distribution in subseqwmt phase unwrapping and phase interpolation, and to control the path of 380
15 February 1994
phase unwrapping. Experimental results for the quantitative determination of the deformation of a skull under loading show that this method can be applied for the analysis of complex phase-shifted holographic interferograms, even if there are local shadows and discontinuities in the fringe pattern.
2. Use of the interferogram intensity modulation distribution for the automated evaluation of complex interference patterns
2.1. Acquisition of phase-shifted holographic interferograms In this paper the attention will be concentrated on the evaluation technique for the phase-shifted holographic interferograms, so that the optical set-up that we used is only briefly described here. Figure 1 gives a schematic overview of this holographic technique. The hologram recording is performed with two well separated reference beams [ 7 ] and two exposures between which state of the object is altered. One of the reference beams serves to record the first object state and is used only in the first exposure, and the other serves to record the second state and is employed only in the second exposure. During reconstruction both reference beams are used simultaneously but one of them has a phase shift by 2 n / N with
Monitor PC Frame grabber
/~
Object
Fig. 1. Experimentalset-upfor recordingand acquisitionof phaseshifted holographic interferograms. Laser: Ar+ laser; BS: beamsplitter; PZT: piezoelectric transducer; M: mirror; L: lens; PT: photothermoplastic camera; CCD: charge-coupleddevice television camera; PC: personal computer.
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respect to the other, where N (N> = 3) is the number of phase shifts employed in the processing. The phase shift is introduced using a piezo-driven mirror. Resulting phase-shifted interferograms are picked-up by a CCD camera and a PC-based frame grabber with 8 bit analog-to-digital converter (ADC) of spatial resolution 512 × 512 pixels. It can be shown that the intensity distribution in the nth phase-shifted interferogram is given by the expression
In(X, Y) =/bias(X, Y) +Imod(X, Y) COS[~(x, y) + 2nn/ N] ,
( 1)
Imod(X, y) =const I T(x, y)12 w~(x, y) w~(x, y), (2) where n = 1, 2, ..., N; I T(x, y) 12 denotes the object reflection coefficient; Wl2, wE are coefficients describing the angular dependence of hologram diffraction efficiency in the first and second exposure, associated with the effect of the recording medium modulation transfer function (MTF), and ~ is the phase function to be measured. Equations ( 1 ), (2) indicate that, in the interferogram areas with a vanishing (due to e.g. surface discontinuities, shadows) object reflection coefficient, a n d / o r with a low (due to the MTF effect) value of w(x, y), Imod vanishes, the interference fringes disappear and the phase becomes uncertain posing serious problems in phase unwrapping.
2.2. Problems in phase unwrapping from the phaseshifted interferograms The principal task of the phase-shifted interferogram analysis is obtaining the phase information O(x, y) from the interference pattern described by eq. ( 1 ). The phase shifting technique employs time as an extra parameter so that the problem is reduced to reading out the phase of a pure sinusoidal signal on the time coordinate, with the space coordinates being fixed at the point to be probed. Using N phase-shifted interferograms the phase function O(x, y) may be retrieved independently of the other parameters in eq. ( 1 ). For the general case of the N-phase algorithm, O(x, y) is expressed as
15 February 1994
Z~=1 In(x, y) sin(2nn/N) O(x'y)=arctanzNn=l In(x,y)cos(2nn/N) "
(3)
By taking the sign of the numerator and denominator into account, the phase values given by eq. (3) can be determined modulo 2n, ranging from - n to n. Therefore, the phase function is wrapped into this range and has discontinuities at + n. In order to obtain a continuous phase function ¢(x, y), one must "unwrap the phase", i.e., add or subtract 2n each time such a jump in phase occurs. The question that needs to be addressed to solve the unwrapping problem is: under what circumstances can the original continuous phase distribution be recovered? If the discrete phase is reliable everywhere, and the maximum phase variation between neighbouring pixels is everywhere less than n, the unwrapping problem is trivial. However the interference patterns obtained in practice, for example in the deformation measurements of the complex objects, contain the phase discontinuities and uncertainties so that the trivial phase unwrapping algorithm does not work. Before starting the phase unwrapping one should solve some problems, which obstruct the unwrapping process. The first problem is to determine the pixels with reliable discrete phase values; those with higher error probability should be excluded from phase unwrapping. The second problem are the phase jumps exceeding n between neighbouring pixels which lead to an incorrect phase unwrapping. To solve these problems, a new approach [ 9] has been proposed that consists of summing unwrapped phase differences over elementary loops comprising four adjacent pixels. If the result is zero everywhere, the phase may be unwrapped along any path. If the result is not zero, it takes either value of + 2n. When the value is 2n, the centre of the elementary loop is a positive pole; when the value is - 2 n , the centre is a negative pole. It is shown that the wrong phase unwrapping occurs along closed paths containing only one pole. The effects of the poles may be eliminated by creating an appropriate linkage between a pair of poles which blocks the path of phase unwrapping. In this sense, we call the linkage between a pair of poles as "block line" in our paper. Thus the third problem to be solved consists in finding the block lines to provide a correct path along which the unwrapping could progress never crossing the pole linkages. 381
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2.3. Intensity modulation distribution
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the reliability of the phase function ~(x, y) and on its discontinuities. We define this parameter, modulation distribution, as 2 ~v ]2
Previous evaluation algorithms are concentrated on retrieving the phase function ~(x, y) from eq. ( 1 ) independent of the parameters Ibm(x, y) and Imod(X, y) involved in this equation. However these parameters may be used to the benefit of the phase unwrapping. In this paper we show that a parameter characterising the intensity modulation of the interference fringes contains important information on
M(x, y) = -- ~[ Z I.(x, y) sin(Znn/N)
NLI_.=
-]2)1/2
+ [ ~= I.(x,y)cos(2nn/N)] ~ .
sampling of N phaseshifted interferograms discrete phase ] calculation i' t
]
discrete intensity ~ modulation calculation
! intensity
modulation l
histogram evaluation
intensity modulation threshold determination
iPrimary mask design i
tithreshold variation i primary mask display and verification
5searching for p o l e s ~ in discrete phase
'L
tracing of block lines I
]interactive correction I
Iphase unwrapping 1
i
Iphase interpolation]
i
Iphase distribution display i
Fig. 2. Flow chartof the modulationanalysisof the phase-shiRedinterfcrograms.
382
(4)
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The substitution of eq. (1) into (4) leads to
M(x, y) --Imod(X, y) .
(5)
Thus M(X, y) has the physical meaning of intensity modulation of the interference fringes. It is well known that speckle noise, image detector noise and quantization introduce errors into the measured intensity In(x, y). As a consequence, a simple relationship exist between M(x, y) and the accuracy of the (calculated) phase function 0(x, y): the lower the modulation distribution, the lower is the accuracy of the phase function. Obviously, the areas of local shadow have a low intensity modulation or it is equal to zero, which means that the phase is uncertain. Furthermore, we observe that in the areas of physical discontinuity the amplitude of the object wave is lower than in the continuous ones due to the difference in the angle of incidence. This leads to a lower intensity modulation as well. When the measured object and a background are located at different distances from the hologram recording medium, the fringe pattern on the background has a lower intensity modulation due to the increase of distance. Hence the modulation distribution can be used for the identification of the area of local shadow, physical discontinuity and background.
2. 4. Modulation analysis of the phase-shifted interferograms Figure 2 shows a flow chart of the modulation analysis of the phase-shifted interferograms. First the discrete phase 0(x, y) and intensity modulation M(x, y) are calculated from Nphase-shifted interferogram data according to eqs. (3) and (4). In analysing the histogram of the intensity modulation distribution a reasonable threshold is introduced to identify valid and invalid pixels. If the threshold is set high, the invalid area on the interferogram increases. This resuits in more reliable phase unwrapping, but also in higher errors in the phase interpolation after the phase unwrapping. If the threshold is set low, the situation is opposite. So we should find a compromise between accuracy and reliability. When a background exists it is easy to set the threshold to separate the background and the measured object correctly. On setting the threshold, a primary binary
15 February 1994
control mask is obtained to separate the valid and invalid pixels. Then in the valid areas we calculate for each pixel the sum of unwrapping phase differences over every elementary square loop consisting of four adjacent pixels to determine the position of positive and negative poles, if those available. Next task is to select the poles to be linked together and to determine the path along which the block lines should be traced. In our algorithm the pixels associated with positive (or negative) poles are the starting points for the tracing of each block line. In the process of tracing the path of the block line from a current pixel to the next one is determined by searching the pixel with a minimum modulation value in the 3 × 3 neighbourhood. When such a pixel is a negative (or positive) pole or belongs to the invalid area, the block line tracing is completed. Thus the linkage formed between a pair of poles generally is not a straight line. The block line tracing can be done automatically in most cases. In our image processing program user-interruption facilities (such as drawing or erasing pole linkages on the two-dimensional distribution of intensity modulation displayed on the monitor) are provided in the event that manual help is necessary to bail the computer out of tricky situations that may happen. After all the poles are linked, the final binary control mask is generated. The valid pixels are binarily marked to be separated from the invalid ones which include the invalid pixels in primary mask, all the poles and the block lines. The phase unwrapping in the valid pixels and then the phase interpolation in the invalid pixels is done separately under the control of the binary mask. This process has been described in detail in refs. [ 10-12 ].
3. Experimental realisation The experimental object is a skull mounted on a holder. The aim of experiment is quantitative evaluation of the deformations of the skull under loading. The value of loading applied between exposures was 0.39N. Since the object has a complex surface shape containing discontinuities and causing shadows, its holographic interferogram presents the complex fringe pattern for which the ordinary phase unwrapping algorithm gives unsatisfactory results. 383
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OPTICS COMMUNICATIONS
Fig. 3. One of five phase-shifted holographic interferograms employed in the evaluation.
15 February 1994
Fig. 5. Two-dimensional grey scale distribution of intensity modulation of the interference fringes. Black: zero modulation.
'?St,;,tt3
,'
ifl~,,t
;!,
rl
, :
,I
r~
U ? j-J'l,, ,,
. . . . . . . . . . . . . . . . .
/it
~. . . . . . . . . . .
Fig. 6. Histogram of the modulation distribution. Fig. 4. Discrete phase distribution. In introducing the phase shift we use the 5-step algorithm to attain a better accuracy [ 13 ]. The phase shifted holographic interferograms picked-up by the C C D camera and the PC-based frame grabber board are stored in the m e m o r y o f the computer system for above described processing. Figure 3 shows one o f the five phase-shifted holographic interferograms. In this figure one can see fringe discontinuities, local 384
shadows and an irregular boundary of the object, which presents an example o f typical situation in the complex object interferogram evaluation. Figure 4 is the discrete phase distribution, in which the phase noise can be clearly seen. Figure 5 is the grey scale two-dimensional distribution o f intensity modulation o f the fringes calculated from the five interferograms according to eq. (4). Comparing fig. 5 with fig. 3 shows that a low intensity modulation
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15 February 1994
Fig. 7. Final control mask showingvalid (grey) and invalid (dark grey) areas, positive (white) and negative (black) poles, and block lines (solid white). exists in the areas o f the background, local shadow, and o f the fringe discontinuities. Figure 6 presents a histogram o f the intensity modulation distribution. In performing the image processing interactively one can move a cursor on the histogram to select different thresholds in the range from 0 to 255, with which the primary binary control mask can be generated. In this test we selected a threshold o f 40. Figure 7 is the final binary control mask. It is seen that the mask covers the shadows and partly the physical discontinuities, and that the measured object is successfully separated from the background. In fig. 7 the location of all poles is shown, for illustration, in the full image. It is evident that most o f them are already cancelled by the control mask and only a few left poles are to be considered. Using our m e t h o d it is not necessary to calculate poles in the full interferogram, but only in its valid areas. Figure 7 shows also the linkages between the positive and negative poles, obtained according to the above described method of tracing m i n i m u m modulation. The tracing was done automatically in our experiment. When fig. 7 is compared with fig. 3, it is apparent that the linkages o f the pole pairs correspond to the actual physical discontinuities. The control mask provides a correct path, along which
Fig. 8. Phase unwrapping progress. (a)-(c) Three different consecutive intermediate stages of automated phase unwrapping under the control of the binary mask. White denotes the unwrapped area of the interferograrn, black the unprocessed area, and grey the invalid area. Note that the phase unwrapping never crosses the phase discontinuities identified by the block lines. 385
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A');
,
~ T ~ T T
b 250" I
1
2¢0
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15 February
1994
a tilt-like deformation clearly seen in fig. 3 - is well reproduced in the calculated phase function as indicated by fig. 9a. In order to evaluate more precisely possible errors we digitally calculated the intensity distributions In (x, y) by substituting into eq. (2) the calculated phase function, Imod(X, y) = M ( x , Y), lbia, (X, Y), the values of the phase shift and compared the results with the corresponding experimentally obtained interferogram intensity distributions. There is a complete fit of these intensity distributions within the valid areas of the discrete phase which means that the calculated phase function and, in particular, the phase discontinuities present in it do not comprise errors. For discussion of the accuracy of phase calculation in phase-shifting interferometry we refer the reader to e.g. ref. [ 14 ]. The experiments show that the evaluation technique presented is powerful enough to cope with the above mentioned problems in quantitative evaluation of holographic interferograms of complex objects.
lOS,~
4. Conclusion
Fig. 9. Calculated phase function which describes the reconstructed deformation field. (a) Pseudo-3Ddisplay of the deformation field. (b) Equal deformation distribution of the deformation field. the following unwrapping progresses never crossing the linkages. Figures 8a-c show three consecutive intermediate stages of the phase unwrapping under the control of the binary mask. The path of the phase unwrapping, as it is clearly seen from fig. 8, successfully goes round the invalid areas and never crosses the block lines, i.e. the phase discontinuities. Figure 9 demonstrates the unwrapped (calculated) phase function which represents the reconstructed deformation field. In fig. 9, (a) is a pseudo3D display of the deformation field, and (b) is the equal deformation distribution of the deformation field. The general shape of the object deformation 386
The modulation analysis of phase-shifted interferograms described in this paper has proven to be effective for the automated evaluation of complex interferograms containing fringe discontinuities. The basis for the proposed evaluation algorithm is a realisation that the intensity modulation of interference fringes, a parameter which commonly has not been taken into account when calculating the discrete phase, contains important information on the accuracy and reliability of the calculated discrete phase, and on the fringe discontinuities, as well as on the boundary between the measured object and the background. The information comprised in the intensity modulation distribution is adequate for constructing a binary mask under whose control the rest of processing, including the phase unwrapping in the valid areas and the phase interpolation in the invalid areas, is done automatically. The binary control mask generated by analysing the histogram of the intensity modulation and tracing the local modulation minima is shown to be an effective means in phase unwrapping.
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C o m p a r e d to the previously d e v e l o p e d m e t h o d s for phase shifted interferogram evaluation, the new evaluation algorithm has the following advantages: ( i ) Easily attainable separation between the measured objects a n d the b a c k g r o u n d through the hist o g r a m o f the intensity m o d u l a t i o n . ( i i ) Decreasing the c o m p u t a t i o n volume, as only a part o f all the poles a n d block lines located in the valid areas o f the p r i m a r y m a s k is u n d e r processing. ( i i i ) Tracing local m o d u l a t i o n m i n i m a assures a good fit o f the block lines to the fringe discontinuities; the block line tracing m a y be done automatically. Experimental results show that this technique can be a p p l i e d for a u t o m a t i n g the evaluation o f complex holographic interferograms, even i f the fringe pattern has discontinuities a n d shaded areas.
References [ 1] G. von Bally, ed., Holography in medicine and biology, Springer Series in Optical Sciences (Springer, Berlin, 1979).
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[2] G. von Bally and P. Greguss, eds., Optics in biomedical sciences, Springer Series in Optical Sciences (Springer, Berlin, 1982). [3]Y.I. Ostrovsky, V.P. Shchepinov and V.V. Yakovlev, Holographic interferometry in experimental mechanics (Springer, Berlin, 1991 ). [4] B. Breuckmann and W. Thieme, Appl. Optics 24 (1985) 2145. [5] T. Kreis, J. Opt. Soc. Am. A 3 (1986) 847. [6] T.A.W.M. Lanen, Optics Comm. 79 (1990) 386. [ 7 ] R. D~indlikerand R. Thalmann, Opt. Eng. 24 ( 1985 ) 824. [8] F. Roddier, Phys. Rep. 170 (1988) 97. [9] D. Barr, V. Coude du Foresto, J. Fox, G.A. Poczulp, J. Richardson, C. Roddier and F. Roddier, Opt. Eng. 30 (1991) 1405. [ 10] D.J. Bone, Appl. Optics 30 ( 1991 ) 3627. [ 11 ] H.A. Vrooman and A.A.M. Maas, Appl. Optics 30 ( 1991 ) 1636. [ 12 ] Xian-Yu Su, G. von Bally and D. Vukicevic, Optics Comm. 98 (1992) 141. [ 13] K.A. Stetson and W.R. Brohinsky, Appl. Optics 24 (1985 ) 3631. Xian-Yu Su, Wen-SenZhou, G. von Bally and D. Vukicevic, Optics Comm. 94 (1992) 561. [ 14 ] K. Kinnst~tter, A.W. Lohmann, J. Schwider and N. Streible, Appl. Optics 27 (1988 ) 5082.
387
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Full length article
Modulation analysis of phase-shifted holographic interferograms X.-Y. Su 1, A.M. Z a r u b i n a n d G. y o n Bally Laboratory of Biophysics, Institute of Experimental Audiology, Universityof Miinster, Kardinal-von-Galen-Ring 10, D-48129 Miinster, Germany Received 7 April 1993; revised manuscript received 20 September 1993
A new evaluation technique for phase-shiftedholographic interferogramsis developed and verified by experiments. In evaluating the interferograms,both the discrete phase distribution and the intensity modulation of interference fringes are calculated and a binary control mask is generatedby analysinghistogramsand tracing local minima of the intensity modulation distribution. The control mask is used to identify valid and invalid areas of the discrete phase distribution in subsequent phase unwrapping and phase interpolation, and to control the path of phase unwrapping. The evaluation of experimentallyobtained phase-shifted holographicinterferograrnsshowsthat this method can be successfullyapplied for the processingof complexinterferencepatterns, even if the interferogramshave fringe discontinuities and shaded areas.
1. Introduction
Holographic interferometry is a powerful technique of deformation and vibration analysis in biomedical and engineering applications [ 1-3 ]. With the development of high resolution CCD camera and the availability of PC-based image grabber, digital holographic interferometry ( D H I ) has become an important diagnostic tool in the quantitative experimental investigations [4-6]. The D H I based on phase shifting technique permits parallel acquisition and processing of large amounts of data with high accuracy, because the phase shifting holographic interferometry utilises a temporal sequence of full-field interferograms of the same phase distribution with fringe shifts within period of the fringes. The phase mapping from digitised N frame fringe data to the wrapped phase is a point to point operation which does not require the analysis of the surrounding points, so there is no problem of spectra overlapping in the spatial frequency domain, which often happens in D H I based Fourier fringe analysis. The later has an obvious advantage that only one interferogram is required, but this advantage can not be rePresent address: Opto-ElectronicsDepartment, Sichuan University, Chengdu 610064, P.R. China.
alised with complex fringe patterns because of the spectra overlapping. When phase-shifting holographic interferometry is applied for deformation measurement of the complex objects, there still are some problems to be solved. First, physical discontinuities of the surface of the complex objects often result in fringe discontinuities in the interferogram. Second, physical discontinuities of the object surface cause local shadows in the interferograms. In the areas of the local shadows the fringes disappear and the phase has an uncertain value. In the areas of the fringe discontinuities the phase changes rapidly. If the phase variation between neighbouring sampling points is more then n, the phase unwrapping problem appears. The third problem is the separation of the measured object from the background. These problems have been the obstacles to the widespread application of holographic interferometry for automated quantitative analysis of the complex objects. In order to control the quality of evaluation of the phase-shifted interferograms and, in particular, to distinguish true interference fringes and other structures such as shadows, contours, holes etc., a proposal has been made [ 7 ] to use the fringe contrast distribution and the mean intensity distribution. However, no algorithm incorporating this proposal
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0030-4018 (93)E0487-Z
379
Volume 105, number 5,6
OPTICS COMMUNICATIONS
and the phase unwrapping ha:. been described, and no experimental results have been presented. The phase unwrapping problem has been extensively studied because of its importance in interferometry and profilometry, but as noted in ref. [8 ], the task of phase unwrapping is difficuL and has been worked on by many people without I5 ~ding an entirely satisfactory solution. Recently, se, reral methods [9-11 ] have been suggested and realised experimentally which cope with interferogran~ evaluation to a great extent. However considerable difficulties still remain when the situation is mare complex. For example, the linkage between a r air of poles [ 9 ] must not be, in general, a straight 1me. Problems appear also if there is an integer fringe shift across the discontinuity, or the spatial frequencies on either side of the discontinuity are not sufficiently different. These situations often occur in the deformation field of the complex objects due to Lhe surface discontinuities. Therefore, in this case a more advanced algorithm is required. In paper [12], a new method of phase-shiffing fringe pattern analysis appliec to profilometry and the experimental results for the 3D shape measurement of a complex object has been described. The purpose of this paper is to de:;cribe the application of the method in D H I and to present experimental results of the complex deforxmtion measurement based on the analysis of hologlaphic interferograms. We introduce into the evaluation process an additional parameter, the intensity modulation of the interference fringes (which is di:;tinct, both physically and mathematically, from the fringe contrast parameter [ 7 ] ), and calculate both lhe discrete phase and intensity modulation distribwJon using N sampled phase-shifted frames of the int,~rference pattern. The following analysis shows thal this parameter contains important information o:a the accuracy and reliability of the discrete p h a s e the physical discontinuities in the fringe pattera, and the boundary between the measured object and the background. By analysing a histogram of the intensity modulation distribution in the whole pat:~ern and tracing local minima of the modulation distribution, a binary control mask is generated. Thq; control mask is used to identify valid and invalid areas of the discrete phase distribution in subseqwmt phase unwrapping and phase interpolation, and to control the path of 380
15 February 1994
phase unwrapping. Experimental results for the quantitative determination of the deformation of a skull under loading show that this method can be applied for the analysis of complex phase-shifted holographic interferograms, even if there are local shadows and discontinuities in the fringe pattern.
2. Use of the interferogram intensity modulation distribution for the automated evaluation of complex interference patterns
2.1. Acquisition of phase-shifted holographic interferograms In this paper the attention will be concentrated on the evaluation technique for the phase-shifted holographic interferograms, so that the optical set-up that we used is only briefly described here. Figure 1 gives a schematic overview of this holographic technique. The hologram recording is performed with two well separated reference beams [ 7 ] and two exposures between which state of the object is altered. One of the reference beams serves to record the first object state and is used only in the first exposure, and the other serves to record the second state and is employed only in the second exposure. During reconstruction both reference beams are used simultaneously but one of them has a phase shift by 2 n / N with
Monitor PC Frame grabber
/~
Object
Fig. 1. Experimentalset-upfor recordingand acquisitionof phaseshifted holographic interferograms. Laser: Ar+ laser; BS: beamsplitter; PZT: piezoelectric transducer; M: mirror; L: lens; PT: photothermoplastic camera; CCD: charge-coupleddevice television camera; PC: personal computer.
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respect to the other, where N (N> = 3) is the number of phase shifts employed in the processing. The phase shift is introduced using a piezo-driven mirror. Resulting phase-shifted interferograms are picked-up by a CCD camera and a PC-based frame grabber with 8 bit analog-to-digital converter (ADC) of spatial resolution 512 × 512 pixels. It can be shown that the intensity distribution in the nth phase-shifted interferogram is given by the expression
In(X, Y) =/bias(X, Y) +Imod(X, Y) COS[~(x, y) + 2nn/ N] ,
( 1)
Imod(X, y) =const I T(x, y)12 w~(x, y) w~(x, y), (2) where n = 1, 2, ..., N; I T(x, y) 12 denotes the object reflection coefficient; Wl2, wE are coefficients describing the angular dependence of hologram diffraction efficiency in the first and second exposure, associated with the effect of the recording medium modulation transfer function (MTF), and ~ is the phase function to be measured. Equations ( 1 ), (2) indicate that, in the interferogram areas with a vanishing (due to e.g. surface discontinuities, shadows) object reflection coefficient, a n d / o r with a low (due to the MTF effect) value of w(x, y), Imod vanishes, the interference fringes disappear and the phase becomes uncertain posing serious problems in phase unwrapping.
2.2. Problems in phase unwrapping from the phaseshifted interferograms The principal task of the phase-shifted interferogram analysis is obtaining the phase information O(x, y) from the interference pattern described by eq. ( 1 ). The phase shifting technique employs time as an extra parameter so that the problem is reduced to reading out the phase of a pure sinusoidal signal on the time coordinate, with the space coordinates being fixed at the point to be probed. Using N phase-shifted interferograms the phase function O(x, y) may be retrieved independently of the other parameters in eq. ( 1 ). For the general case of the N-phase algorithm, O(x, y) is expressed as
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Z~=1 In(x, y) sin(2nn/N) O(x'y)=arctanzNn=l In(x,y)cos(2nn/N) "
(3)
By taking the sign of the numerator and denominator into account, the phase values given by eq. (3) can be determined modulo 2n, ranging from - n to n. Therefore, the phase function is wrapped into this range and has discontinuities at + n. In order to obtain a continuous phase function ¢(x, y), one must "unwrap the phase", i.e., add or subtract 2n each time such a jump in phase occurs. The question that needs to be addressed to solve the unwrapping problem is: under what circumstances can the original continuous phase distribution be recovered? If the discrete phase is reliable everywhere, and the maximum phase variation between neighbouring pixels is everywhere less than n, the unwrapping problem is trivial. However the interference patterns obtained in practice, for example in the deformation measurements of the complex objects, contain the phase discontinuities and uncertainties so that the trivial phase unwrapping algorithm does not work. Before starting the phase unwrapping one should solve some problems, which obstruct the unwrapping process. The first problem is to determine the pixels with reliable discrete phase values; those with higher error probability should be excluded from phase unwrapping. The second problem are the phase jumps exceeding n between neighbouring pixels which lead to an incorrect phase unwrapping. To solve these problems, a new approach [ 9] has been proposed that consists of summing unwrapped phase differences over elementary loops comprising four adjacent pixels. If the result is zero everywhere, the phase may be unwrapped along any path. If the result is not zero, it takes either value of + 2n. When the value is 2n, the centre of the elementary loop is a positive pole; when the value is - 2 n , the centre is a negative pole. It is shown that the wrong phase unwrapping occurs along closed paths containing only one pole. The effects of the poles may be eliminated by creating an appropriate linkage between a pair of poles which blocks the path of phase unwrapping. In this sense, we call the linkage between a pair of poles as "block line" in our paper. Thus the third problem to be solved consists in finding the block lines to provide a correct path along which the unwrapping could progress never crossing the pole linkages. 381
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2.3. Intensity modulation distribution
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the reliability of the phase function ~(x, y) and on its discontinuities. We define this parameter, modulation distribution, as 2 ~v ]2
Previous evaluation algorithms are concentrated on retrieving the phase function ~(x, y) from eq. ( 1 ) independent of the parameters Ibm(x, y) and Imod(X, y) involved in this equation. However these parameters may be used to the benefit of the phase unwrapping. In this paper we show that a parameter characterising the intensity modulation of the interference fringes contains important information on
M(x, y) = -- ~[ Z I.(x, y) sin(Znn/N)
NLI_.=
-]2)1/2
+ [ ~= I.(x,y)cos(2nn/N)] ~ .
sampling of N phaseshifted interferograms discrete phase ] calculation i' t
]
discrete intensity ~ modulation calculation
! intensity
modulation l
histogram evaluation
intensity modulation threshold determination
iPrimary mask design i
tithreshold variation i primary mask display and verification
5searching for p o l e s ~ in discrete phase
'L
tracing of block lines I
]interactive correction I
Iphase unwrapping 1
i
Iphase interpolation]
i
Iphase distribution display i
Fig. 2. Flow chartof the modulationanalysisof the phase-shiRedinterfcrograms.
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(4)
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The substitution of eq. (1) into (4) leads to
M(x, y) --Imod(X, y) .
(5)
Thus M(X, y) has the physical meaning of intensity modulation of the interference fringes. It is well known that speckle noise, image detector noise and quantization introduce errors into the measured intensity In(x, y). As a consequence, a simple relationship exist between M(x, y) and the accuracy of the (calculated) phase function 0(x, y): the lower the modulation distribution, the lower is the accuracy of the phase function. Obviously, the areas of local shadow have a low intensity modulation or it is equal to zero, which means that the phase is uncertain. Furthermore, we observe that in the areas of physical discontinuity the amplitude of the object wave is lower than in the continuous ones due to the difference in the angle of incidence. This leads to a lower intensity modulation as well. When the measured object and a background are located at different distances from the hologram recording medium, the fringe pattern on the background has a lower intensity modulation due to the increase of distance. Hence the modulation distribution can be used for the identification of the area of local shadow, physical discontinuity and background.
2. 4. Modulation analysis of the phase-shifted interferograms Figure 2 shows a flow chart of the modulation analysis of the phase-shifted interferograms. First the discrete phase 0(x, y) and intensity modulation M(x, y) are calculated from Nphase-shifted interferogram data according to eqs. (3) and (4). In analysing the histogram of the intensity modulation distribution a reasonable threshold is introduced to identify valid and invalid pixels. If the threshold is set high, the invalid area on the interferogram increases. This resuits in more reliable phase unwrapping, but also in higher errors in the phase interpolation after the phase unwrapping. If the threshold is set low, the situation is opposite. So we should find a compromise between accuracy and reliability. When a background exists it is easy to set the threshold to separate the background and the measured object correctly. On setting the threshold, a primary binary
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control mask is obtained to separate the valid and invalid pixels. Then in the valid areas we calculate for each pixel the sum of unwrapping phase differences over every elementary square loop consisting of four adjacent pixels to determine the position of positive and negative poles, if those available. Next task is to select the poles to be linked together and to determine the path along which the block lines should be traced. In our algorithm the pixels associated with positive (or negative) poles are the starting points for the tracing of each block line. In the process of tracing the path of the block line from a current pixel to the next one is determined by searching the pixel with a minimum modulation value in the 3 × 3 neighbourhood. When such a pixel is a negative (or positive) pole or belongs to the invalid area, the block line tracing is completed. Thus the linkage formed between a pair of poles generally is not a straight line. The block line tracing can be done automatically in most cases. In our image processing program user-interruption facilities (such as drawing or erasing pole linkages on the two-dimensional distribution of intensity modulation displayed on the monitor) are provided in the event that manual help is necessary to bail the computer out of tricky situations that may happen. After all the poles are linked, the final binary control mask is generated. The valid pixels are binarily marked to be separated from the invalid ones which include the invalid pixels in primary mask, all the poles and the block lines. The phase unwrapping in the valid pixels and then the phase interpolation in the invalid pixels is done separately under the control of the binary mask. This process has been described in detail in refs. [ 10-12 ].
3. Experimental realisation The experimental object is a skull mounted on a holder. The aim of experiment is quantitative evaluation of the deformations of the skull under loading. The value of loading applied between exposures was 0.39N. Since the object has a complex surface shape containing discontinuities and causing shadows, its holographic interferogram presents the complex fringe pattern for which the ordinary phase unwrapping algorithm gives unsatisfactory results. 383
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Fig. 3. One of five phase-shifted holographic interferograms employed in the evaluation.
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Fig. 5. Two-dimensional grey scale distribution of intensity modulation of the interference fringes. Black: zero modulation.
'?St,;,tt3
,'
ifl~,,t
;!,
rl
, :
,I
r~
U ? j-J'l,, ,,
. . . . . . . . . . . . . . . . .
/it
~. . . . . . . . . . .
Fig. 6. Histogram of the modulation distribution. Fig. 4. Discrete phase distribution. In introducing the phase shift we use the 5-step algorithm to attain a better accuracy [ 13 ]. The phase shifted holographic interferograms picked-up by the C C D camera and the PC-based frame grabber board are stored in the m e m o r y o f the computer system for above described processing. Figure 3 shows one o f the five phase-shifted holographic interferograms. In this figure one can see fringe discontinuities, local 384
shadows and an irregular boundary of the object, which presents an example o f typical situation in the complex object interferogram evaluation. Figure 4 is the discrete phase distribution, in which the phase noise can be clearly seen. Figure 5 is the grey scale two-dimensional distribution o f intensity modulation o f the fringes calculated from the five interferograms according to eq. (4). Comparing fig. 5 with fig. 3 shows that a low intensity modulation
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Fig. 7. Final control mask showingvalid (grey) and invalid (dark grey) areas, positive (white) and negative (black) poles, and block lines (solid white). exists in the areas o f the background, local shadow, and o f the fringe discontinuities. Figure 6 presents a histogram o f the intensity modulation distribution. In performing the image processing interactively one can move a cursor on the histogram to select different thresholds in the range from 0 to 255, with which the primary binary control mask can be generated. In this test we selected a threshold o f 40. Figure 7 is the final binary control mask. It is seen that the mask covers the shadows and partly the physical discontinuities, and that the measured object is successfully separated from the background. In fig. 7 the location of all poles is shown, for illustration, in the full image. It is evident that most o f them are already cancelled by the control mask and only a few left poles are to be considered. Using our m e t h o d it is not necessary to calculate poles in the full interferogram, but only in its valid areas. Figure 7 shows also the linkages between the positive and negative poles, obtained according to the above described method of tracing m i n i m u m modulation. The tracing was done automatically in our experiment. When fig. 7 is compared with fig. 3, it is apparent that the linkages o f the pole pairs correspond to the actual physical discontinuities. The control mask provides a correct path, along which
Fig. 8. Phase unwrapping progress. (a)-(c) Three different consecutive intermediate stages of automated phase unwrapping under the control of the binary mask. White denotes the unwrapped area of the interferograrn, black the unprocessed area, and grey the invalid area. Note that the phase unwrapping never crosses the phase discontinuities identified by the block lines. 385
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A');
,
~ T ~ T T
b 250" I
1
2¢0
]50
i
f
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1994
a tilt-like deformation clearly seen in fig. 3 - is well reproduced in the calculated phase function as indicated by fig. 9a. In order to evaluate more precisely possible errors we digitally calculated the intensity distributions In (x, y) by substituting into eq. (2) the calculated phase function, Imod(X, y) = M ( x , Y), lbia, (X, Y), the values of the phase shift and compared the results with the corresponding experimentally obtained interferogram intensity distributions. There is a complete fit of these intensity distributions within the valid areas of the discrete phase which means that the calculated phase function and, in particular, the phase discontinuities present in it do not comprise errors. For discussion of the accuracy of phase calculation in phase-shifting interferometry we refer the reader to e.g. ref. [ 14 ]. The experiments show that the evaluation technique presented is powerful enough to cope with the above mentioned problems in quantitative evaluation of holographic interferograms of complex objects.
lOS,~
4. Conclusion
Fig. 9. Calculated phase function which describes the reconstructed deformation field. (a) Pseudo-3Ddisplay of the deformation field. (b) Equal deformation distribution of the deformation field. the following unwrapping progresses never crossing the linkages. Figures 8a-c show three consecutive intermediate stages of the phase unwrapping under the control of the binary mask. The path of the phase unwrapping, as it is clearly seen from fig. 8, successfully goes round the invalid areas and never crosses the block lines, i.e. the phase discontinuities. Figure 9 demonstrates the unwrapped (calculated) phase function which represents the reconstructed deformation field. In fig. 9, (a) is a pseudo3D display of the deformation field, and (b) is the equal deformation distribution of the deformation field. The general shape of the object deformation 386
The modulation analysis of phase-shifted interferograms described in this paper has proven to be effective for the automated evaluation of complex interferograms containing fringe discontinuities. The basis for the proposed evaluation algorithm is a realisation that the intensity modulation of interference fringes, a parameter which commonly has not been taken into account when calculating the discrete phase, contains important information on the accuracy and reliability of the calculated discrete phase, and on the fringe discontinuities, as well as on the boundary between the measured object and the background. The information comprised in the intensity modulation distribution is adequate for constructing a binary mask under whose control the rest of processing, including the phase unwrapping in the valid areas and the phase interpolation in the invalid areas, is done automatically. The binary control mask generated by analysing the histogram of the intensity modulation and tracing the local modulation minima is shown to be an effective means in phase unwrapping.
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C o m p a r e d to the previously d e v e l o p e d m e t h o d s for phase shifted interferogram evaluation, the new evaluation algorithm has the following advantages: ( i ) Easily attainable separation between the measured objects a n d the b a c k g r o u n d through the hist o g r a m o f the intensity m o d u l a t i o n . ( i i ) Decreasing the c o m p u t a t i o n volume, as only a part o f all the poles a n d block lines located in the valid areas o f the p r i m a r y m a s k is u n d e r processing. ( i i i ) Tracing local m o d u l a t i o n m i n i m a assures a good fit o f the block lines to the fringe discontinuities; the block line tracing m a y be done automatically. Experimental results show that this technique can be a p p l i e d for a u t o m a t i n g the evaluation o f complex holographic interferograms, even i f the fringe pattern has discontinuities a n d shaded areas.
References [ 1] G. von Bally, ed., Holography in medicine and biology, Springer Series in Optical Sciences (Springer, Berlin, 1979).
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[2] G. von Bally and P. Greguss, eds., Optics in biomedical sciences, Springer Series in Optical Sciences (Springer, Berlin, 1982). [3]Y.I. Ostrovsky, V.P. Shchepinov and V.V. Yakovlev, Holographic interferometry in experimental mechanics (Springer, Berlin, 1991 ). [4] B. Breuckmann and W. Thieme, Appl. Optics 24 (1985) 2145. [5] T. Kreis, J. Opt. Soc. Am. A 3 (1986) 847. [6] T.A.W.M. Lanen, Optics Comm. 79 (1990) 386. [ 7 ] R. D~indlikerand R. Thalmann, Opt. Eng. 24 ( 1985 ) 824. [8] F. Roddier, Phys. Rep. 170 (1988) 97. [9] D. Barr, V. Coude du Foresto, J. Fox, G.A. Poczulp, J. Richardson, C. Roddier and F. Roddier, Opt. Eng. 30 (1991) 1405. [ 10] D.J. Bone, Appl. Optics 30 ( 1991 ) 3627. [ 11 ] H.A. Vrooman and A.A.M. Maas, Appl. Optics 30 ( 1991 ) 1636. [ 12 ] Xian-Yu Su, G. von Bally and D. Vukicevic, Optics Comm. 98 (1992) 141. [ 13] K.A. Stetson and W.R. Brohinsky, Appl. Optics 24 (1985 ) 3631. Xian-Yu Su, Wen-SenZhou, G. von Bally and D. Vukicevic, Optics Comm. 94 (1992) 561. [ 14 ] K. Kinnst~tter, A.W. Lohmann, J. Schwider and N. Streible, Appl. Optics 27 (1988 ) 5082.
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